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Mirrors > Home > MPE Home > Th. List > oawordex | Structured version Visualization version GIF version |
Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59 and its converse. See oawordeu 8582 for uniqueness. (Contributed by NM, 12-Dec-2004.) |
Ref | Expression |
---|---|
oawordex | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oawordeu 8582 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) | |
2 | 1 | ex 411 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) |
3 | reurex 3378 | . . 3 ⊢ (∃!𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) | |
4 | 2, 3 | syl6 35 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) |
5 | oawordexr 8583 | . . . 4 ⊢ ((𝐴 ∈ On ∧ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) → 𝐴 ⊆ 𝐵) | |
6 | 5 | ex 411 | . . 3 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵)) |
7 | 6 | adantr 479 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵)) |
8 | 4, 7 | impbid 211 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 ∃!wreu 3372 ⊆ wss 3949 Oncon0 6374 (class class class)co 7426 +o coa 8490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-oadd 8497 |
This theorem is referenced by: oaordex 8585 oaass 8588 odi 8606 omeulem1 8609 oasubex 42746 oaabsb 42754 oawordex2 42786 oawordex3 42861 |
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