Mathbox for Rohan Ridenour |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > rr-phpd | Structured version Visualization version GIF version |
Description: Equivalent of php 8701 without negation. (Contributed by Rohan Ridenour, 3-Aug-2023.) |
Ref | Expression |
---|---|
rr-phpd.1 | ⊢ (𝜑 → 𝐴 ∈ ω) |
rr-phpd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
rr-phpd.3 | ⊢ (𝜑 → 𝐴 ≈ 𝐵) |
Ref | Expression |
---|---|
rr-phpd | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rr-phpd.3 | . 2 ⊢ (𝜑 → 𝐴 ≈ 𝐵) | |
2 | rr-phpd.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ω) | |
3 | rr-phpd.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝐴) | |
4 | 3 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ⊆ 𝐴) |
5 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
6 | 5 | neqcomd 2831 | . . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐵 = 𝐴) |
7 | dfpss2 4062 | . . . . 5 ⊢ (𝐵 ⊊ 𝐴 ↔ (𝐵 ⊆ 𝐴 ∧ ¬ 𝐵 = 𝐴)) | |
8 | 4, 6, 7 | sylanbrc 585 | . . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ⊊ 𝐴) |
9 | php 8701 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ¬ 𝐴 ≈ 𝐵) | |
10 | 2, 8, 9 | syl2an2r 683 | . . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 ≈ 𝐵) |
11 | 10 | ex 415 | . 2 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → ¬ 𝐴 ≈ 𝐵)) |
12 | 1, 11 | mt4d 117 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ⊊ wpss 3937 class class class wbr 5066 ωcom 7580 ≈ cen 8506 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-om 7581 df-er 8289 df-en 8510 df-dom 8511 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |