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Mirrors > Home > MPE Home > Th. List > infeq5i | Structured version Visualization version GIF version |
Description: Half of infeq5 9632. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
infeq5i | ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 5328 | . 2 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V) | |
2 | 0ex 5308 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | snid 4665 | . . . 4 ⊢ ∅ ∈ {∅} |
4 | disj4 4459 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω) | |
5 | disj3 4454 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅})) | |
6 | 4, 5 | bitr3i 277 | . . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅})) |
7 | peano1 7879 | . . . . . . 7 ⊢ ∅ ∈ ω | |
8 | eleq2 2823 | . . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅}))) | |
9 | 7, 8 | mpbii 232 | . . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅})) |
10 | 9 | eldifbd 3962 | . . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅}) |
11 | 6, 10 | sylbi 216 | . . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅}) |
12 | 3, 11 | mt4 116 | . . 3 ⊢ (ω ∖ {∅}) ⊊ ω |
13 | unidif0 5359 | . . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω | |
14 | limom 7871 | . . . . . 6 ⊢ Lim ω | |
15 | limuni 6426 | . . . . . 6 ⊢ (Lim ω → ω = ∪ ω) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ ω = ∪ ω |
17 | 13, 16 | eqtr4i 2764 | . . . 4 ⊢ ∪ (ω ∖ {∅}) = ω |
18 | 17 | psseq2i 4091 | . . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω) |
19 | 12, 18 | mpbir 230 | . 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) |
20 | psseq1 4088 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥)) | |
21 | unieq 4920 | . . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅})) | |
22 | 21 | psseq2d 4094 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
23 | 20, 22 | bitrd 279 | . . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
24 | 23 | spcegv 3588 | . 2 ⊢ ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) → ∃𝑥 𝑥 ⊊ ∪ 𝑥)) |
25 | 1, 19, 24 | mpisyl 21 | 1 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3946 ∩ cin 3948 ⊊ wpss 3950 ∅c0 4323 {csn 4629 ∪ cuni 4909 Lim wlim 6366 ωcom 7855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-om 7856 |
This theorem is referenced by: infeq5 9632 inf5 9640 |
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